Pole-zero analysis for stability in Signal Processing

Introduction

Pole-zero analysis helps us understand if a system will behave well or become unstable. It shows where the system's key points (poles and zeros) are on a graph.

Checking if an audio filter will produce clear sound without noise.

Making sure a control system in a robot moves smoothly without shaking.

Designing a signal filter that removes unwanted signals but stays stable.

Testing if an electronic circuit will work safely without oscillations.

Syntax

Signal Processing

H(z) = (z - zero1)(z - zero2)... / (z - pole1)(z - pole2)...

Poles are values of z that make the denominator zero.

Zeros are values of z that make the numerator zero.

Examples

One zero at 0.5 and one pole at 0.9.

Signal Processing

H(z) = (z - 0.5) / (z - 0.9)

Two zeros at 0.3 and -0.4, two poles at 0.8 and 0.7.

Signal Processing

H(z) = (z - 0.3)(z + 0.4) / (z - 0.8)(z - 0.7)

Sample Program

This code plots poles and zeros on the complex plane with the unit circle. It then checks if all poles are inside the unit circle to decide stability.

Signal Processing

import numpy as np
import matplotlib.pyplot as plt

# Define poles and zeros
zeros = [0.5, -0.4]
poles = [0.8, 0.7]

# Plot unit circle
theta = np.linspace(0, 2*np.pi, 400)
x = np.cos(theta)
y = np.sin(theta)

plt.figure(figsize=(6,6))
plt.plot(x, y, 'k--', label='Unit Circle')

# Plot zeros
plt.scatter(np.real(zeros), np.imag(zeros), marker='o', facecolors='none', edgecolors='b', s=100, label='Zeros')
# Plot poles
plt.scatter(np.real(poles), np.imag(poles), marker='x', color='r', s=100, label='Poles')

plt.title('Pole-Zero Plot for Stability Analysis')
plt.xlabel('Real Part')
plt.ylabel('Imaginary Part')
plt.grid(True)
plt.axis('equal')
plt.legend()
plt.show()

# Check stability: all poles must be inside unit circle
stable = all(abs(p) < 1 for p in poles)
print(f"System is {'stable' if stable else 'unstable'}.")

Important Notes

For discrete systems, poles inside the unit circle mean stability.

Zeros affect the system's frequency response but not stability directly.

Poles on or outside the unit circle mean the system can become unstable.

Summary

Pole-zero plots show key points of a system on the complex plane.

Stability depends on where poles lie relative to the unit circle.

Plotting helps visualize and decide if a system will behave well.